# Disconnected linear system of divisors?

On a non-singular projective variety, the complete linear system $$|D|$$ of a divisor $$D$$ is the set of all effective divisors linearly equivalent to $$D$$. Often we speak of the dimension $$\mathrm{dim}|D|$$, which is the number of parameters.

Consider in particular a complex projective variety, with an effective divisor $$D$$. Can there be disconnected components of $$|D|$$? For example, can it be that $$\mathrm{dim}|D|=0$$ while there are multiple elements in $$|D|$$?

Personally I would be particularly interested in examples of divisors $$D$$ on a complex surface where the self-intersection is negative, $$D^2 < 0$$: here $$\mathrm{dim}|D|=0$$ but I wonder if there may be several elements in $$|D|$$.

• Remember that $|D|$ is a projective space, namely, $\Bbb P(H^0(\mathscr O(D)))$. – Ted Shifrin Mar 24 at 0:41
• @TedShifrin Thanks a lot for your comment; this makes it totally clear. I will delete this question as I don't think it is useful for anyone else. – diracula Mar 24 at 0:49
• Don't delete. I thought it was a good question. Perhaps I should turn my comment into an answer. – Ted Shifrin Mar 24 at 0:59

In fact, $$|D|$$ is a projective space — namely, $$\Bbb P(H^0(D)))$$. The linear system is coming from "continuous" variation of the divisor, as it consists of divisors that are linearly equivalent to the given divisor. Linear equivalence, in particular, is a specific sort of homotopy, and the divisors can be "connected" in a continuous way.